TY - GEN
T1 - On non-unique solutions in mean field games
AU - Hajek, Bruce
AU - Livesay, Michael
N1 - Publisher Copyright:
© 2019 IEEE.
PY - 2019/12
Y1 - 2019/12
N2 - The theory of mean field games is a tool to understand noncooperative dynamic stochastic games with a large number of players. Much of the theory has evolved under conditions ensuring uniqueness of the mean field game Nash equilibrium. However, in some situations, typically involving symmetry breaking, non-uniqueness of solutions is an essential feature. To investigate the nature of non-unique solutions, this paper focuses on the technically simple setting where players have one of two states, with continuous time dynamics, and the game is symmetric in the players, and players are restricted to using Markov strategies. All the mean field game Nash equilibria are identified for a symmetric follow the crowd game. Such equilibria correspond to symmetric ϵ-Nash Markov equilibria for N players with ϵ converging to zero as N goes to infinity.In contrast to the mean field game, there is a unique Nash equilibrium for finite N. It is shown that fluid limits arising from the Nash equilibria for finite N as N goes to infinity are mean field game Nash equilibria, and evidence is given supporting the conjecture that such limits, among all mean field game Nash equilibria, are the ones that are stable fixed points of the mean field best response mapping.
AB - The theory of mean field games is a tool to understand noncooperative dynamic stochastic games with a large number of players. Much of the theory has evolved under conditions ensuring uniqueness of the mean field game Nash equilibrium. However, in some situations, typically involving symmetry breaking, non-uniqueness of solutions is an essential feature. To investigate the nature of non-unique solutions, this paper focuses on the technically simple setting where players have one of two states, with continuous time dynamics, and the game is symmetric in the players, and players are restricted to using Markov strategies. All the mean field game Nash equilibria are identified for a symmetric follow the crowd game. Such equilibria correspond to symmetric ϵ-Nash Markov equilibria for N players with ϵ converging to zero as N goes to infinity.In contrast to the mean field game, there is a unique Nash equilibrium for finite N. It is shown that fluid limits arising from the Nash equilibria for finite N as N goes to infinity are mean field game Nash equilibria, and evidence is given supporting the conjecture that such limits, among all mean field game Nash equilibria, are the ones that are stable fixed points of the mean field best response mapping.
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U2 - 10.1109/CDC40024.2019.9029906
DO - 10.1109/CDC40024.2019.9029906
M3 - Conference contribution
AN - SCOPUS:85082473712
T3 - Proceedings of the IEEE Conference on Decision and Control
SP - 1219
EP - 1224
BT - 2019 IEEE 58th Conference on Decision and Control, CDC 2019
PB - Institute of Electrical and Electronics Engineers Inc.
T2 - 58th IEEE Conference on Decision and Control, CDC 2019
Y2 - 11 December 2019 through 13 December 2019
ER -